Algebra and Logic

, Volume 45, Issue 4, pp 220–231 | Cite as

Recognition of finite groups by the prime graph

  • A. V. Zavarnitsine


We obtain the first example of an infinite series of finite simple groups that are uniquely determined by their prime graph in the class of all finite groups. We also show that there exist almost simple groups for which the number of finite groups with the same prime graph is equal to 2.


prime graph finite group modular representations recognition 


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  1. 1.
    V. D. Mazurov, “Groups with prescribed orders of elements,” Izv. Ural. Gos. Univ., Mat. Mekh., 36, No. 7, 119–138 (2005).MathSciNetGoogle Scholar
  2. 2.
    A. Khosravi and B. Khosravi, “Quasirecognition of the simple group 2G2(q) by the prime graph,” to appear in Sib. Mat. Zh. Google Scholar
  3. 3.
    J. S. Williams, “Prime graph components of finite groups,” J. Alg., 69, No. 2, 487–513 (1981).zbMATHCrossRefGoogle Scholar
  4. 4.
    A. S. Kondratiev, “On prime graph components for finite simple groups,” Mat. Sb., 180, No. 6, 787–797 (1989).Google Scholar
  5. 5.
    W. Shi, “The characterization of the sporadic simple groups by their element orders,” Alg. Coll., 1, No. 2, 159–166 (1994).zbMATHGoogle Scholar
  6. 6.
    A. V. Zavarnitsine, “Recognition of the simple groups L 3(q) by element orders,” J. Group Theory, 7, No. 1, 81–97 (2004).zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    V. D. Mazurov, M. C. Xu, and H. P. Cao, “Recognition of finite simple groups L 3(2m) and U 3(2m) by their element orders,” Algebra Logika, 39, No. 5, 567–585 (2000).zbMATHGoogle Scholar
  8. 8.
    V. D. Mazurov, “Recognition of finite simple groups S 4(q) by their element orders,” Algebra Logika, 41, No. 2, 166–198 (2002).zbMATHMathSciNetGoogle Scholar
  9. 9.
    V. D. Mazurov, “The set of orders of elements in a finite group,” Algebra Logika, 33, No. 1, 81–89 (1994).zbMATHMathSciNetGoogle Scholar
  10. 10.
    K. Zsigmondy, “Zur Theorie der Potenzreste,” Mon. Math. Phys., 3, 265–284 (1892).zbMATHCrossRefGoogle Scholar
  11. 11.
    The GAP Group, GAP — Groups, Algorithms, and Programming, Vers. 4.4.7 (2006); http: // Scholar
  12. 12.
    R. M. Guralnick and P. H. Tiep, “Finite simple unisingular groups of Lie type,” J. Group Theory, 6, No. 3, 271–310 (2003).zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    B. Chang and R. Ree, “The character of G 2(q),” in Symp. Math., Vol. 13, Academic Press, London (1974), pp. 395–413.Google Scholar
  14. 14.
    G. Hiss and J. Shamash, “3-blocks and 3-modular characters of G 2(q),” J. Alg., 131, No. 2, 371–387 (1990).zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    H. N. Ward, “On Ree’s series of simple groups,” Trans. Am. Math. Soc., 121, No. 1, 62–89 (1966).zbMATHCrossRefGoogle Scholar
  16. 16.
    P. Landrock and G. O. Michler, “Principal 2-blocks of the simple groups of Ree type,” Trans. Am. Math. Soc., 260, 83–111 (1980).zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    B. Huppert and N. Blackburn, Finite Groups, Vol. 3, Springer, Berlin (1982).Google Scholar
  18. 18.
    J. Conway, R. Curtis, S. Norton, et al., Atlas of Finite Groups, Clarendon, Oxford (1985).zbMATHGoogle Scholar
  19. 19.
    M. Hagie, “The prime graph of a sporadic simple group,” Comm. Alg., 31, No. 9, 4405–4424 (2003).zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • A. V. Zavarnitsine
    • 1
  1. 1.Institute of Mathematics, Siberian BranchRussian Academy of SciencesNovosibirskRussia

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